domain: all people
$F(x, y) ≡ x$ and $y$ are friends
$I(x) ≡ x$ is a football player.
$$∃x∀y(F(x, y) → ¬I(y))$$
there exists a person $x$, for every person $y$, if $x$ and $y$ are friends then $x$ is not a football player. I need to translate this to "natural" English.
"There is a person that is a friend with every person whose friends are not a football player." when I say this, I feel that this sounds like $∃x∀y(F(x, y) ∧ ¬I(y))$.
Would this be correct? Can this be further simply expressed?
On
"There is a person, whose all of his friends (if he has any) are not football players",
"There is a person, whose none of his friends (if he has any) are football players", or
"There is a person, if he has any friends, none of them is a football player".
In more details $$∃x∀y(F(x, y) → ¬I(y))$$ says:
There is a person $x$, whenever some other person $y$ is his friend, that other person $y$ is not a football player.
On
No, your translation is not correct for the reasons explained by @Bram28. Following are two English sentences translated from your formula:
∃x∀y(¬F(x,y) ∨ ¬I(y)) is an equivalent formula. This could be translated into:
∃x∀y(I(y) → ¬F(x,y)) is another equivalent formula. This could be translated to:
No. Notice how your "There is a person that is a friend with every person whose friends are not a football player." contains two 'friend' relationships. Indeed, your sentence would be symbolized in logic as:
$\exists x \forall y (\forall z (F(y,z) \to \neg I(z)) \to F(x,y))$
So that's a good bit more complicated than the logic sentence that's given to you ... although that first part is very similar .. and that should give you a clue how to translate your logic sentence.